Slope of a Line

Q.

Let three vectors $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ be such that $\overrightarrow{c}$ is coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$, $\overrightarrow{a}·\overrightarrow{c}=7$ and $\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$ where $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}+\hat{k}$ then the value of $2{\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|}^2$ is _______.

Q.

Two sides of a rhombus are along the lines x-y+1=0 and 7x-y-5=0. If its diagonals intersect at (-1, -2), then which one of the following is a vertex of this rhombus?

1. $(-3,-9)$

2. $(-3,-8)$

3. $(\frac{1}{3},-\frac{8}{3})$

4. $(-\frac{10}{3},-\frac{7}{3})$

Q.

What can be said regarding a line if its slope is

(i) zero (ii) positive (iii) negative ?

Q.

If the curve $y=a{x}^2+bx+c;x\in R$ passes through the point $(1,2)$ and the tangent line to this curve at origin is $y=x$, then the possible values of $a,b,c$ are:

1. $a=1,b=1,c=0$

2. $a=-1,b=1,c=1$

3. $a=1,b=0,c=1$

4. $a=\frac{1}{2},b=\frac{1}{2},c=1$

Q. The graph of $y=x\ln x$ is

1. 
2. 
3. 
4. 

Q.

Find the slopes of the lines which make the following angles with the positive direction of x-axis:

$\left(i\right)-\frac{\pi }{4}\left(ii\right)\frac{2\pi }{3}\left(iii\right)\frac{3\pi }{4}\left(iv\right)\frac{\pi }{3}$

Q. The points $(0,\frac{8}{3}),(1,3)$ and $(82,30)$:

1. form an acute angled triangle.
2. lie on a straight line.
3. form an obtuse angled triangle
4. form a right angled triangle.

Q. If $m_1\&m_2$ are the slopes of two intersecting lines $(m_1.m_2\ne -1)\&\theta$ is the acute angle between them, then $tan\theta =|\frac{m_1+m_2}{1-m_1.m_2}|$

1. True

2. False

Q.

A $3^{\mathrm{rd}}$-degree binomial with a constant term of $8$.

Q. A rectangle $ABCD$ has its side $AB$ parallel to line $y=x$ and vertices $A,B$ and $D$ lie on $y=1,x=2$ and $x=-2$ respectively. Locus of vertex ${}^{\prime }{C}^{\prime }$ is

1. $y=5$
2. $x-y=5$
3. $x=5$
4. $x+y=5$

Q. If the straight line $3x+4y+5-k(x+y+3)=0$ is parallel to $y-$ axis, then the value of $k$ is

Q.

Find the slope of a line (i) which bisects the first quadrant angle (ii) which makes an angle of ${30}^{\circ }$ with the positive direction of y-axis measured anticlockwise.

Q. The locus of the mid-point of the chord of a circle $x^2+y^2=4$ such that the segment intercepted by the chord on the curve $x^2-2x–2y=0$ subtends a right angle at the origin is

1. $x^2+y^2-2x-2y=0$
2. $x^2+y^2+2x-2y=0$
3. $x^2+y^2+2x+2y=0$
4. $x^2+y^2-2x+2y=0$

Q. A circle of radius 'r' is concentric with the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Then inclination of common tangent with major axis is ___ (b<r<a)

1. $ta{n}^{-1}\left(\frac{r^2-b^2}{a^2-r^2}\right)$
2. $\frac{\pi }{2}$
3. $ta{n}^{-1}\left(\frac{b}{a}\right)$
4. $ta{n}^{-1}\left(\frac{rb}{a}\right)$

Q. Find the slope of the line perpendicular to AB if:

$\left(i\right)A=(0,-5)\text{and}B=(-2,4)$

Q. If the vertices of a quadrilateral $ABCD$ are $A(-4,2),B(2,6),C(8,5)$ and $D(9,-7)$, then the quadrilateral formed by joining the midpoints of the sides of the quadrilateral $ABCD$ will be a

1. Parallelogram
2. Rectangle
3. Square
4. Rhombus

Q.

What are two lines that are perpendicular?

Q. The real value of $\lambda$ for which the image of the point $(\lambda ,\lambda -1)$ with respect to the line $3x+y=6\lambda$ is the point $({\lambda }^2+1,\lambda )$ is

1. $-1$
2. $1$
3. $0$
4. $2$

Q. The graphs $y=2x^3-4x+2$ and $y=x^3+2x-1$ intersect at exactly three distinct points. The slope of the line passing through two of these points

1. is equal to $4$
2. is equal to $6$
3. is equal to $8$
4. is not unique

Q.

The equation of the line parallel to $x$ axis and tangent to the curve $y=\frac{1}{x^2+2x+5}$ is

1. $y=\frac{1}{4}$

2. $y=4$

3. $y=\frac{1}{2}$

4. $y=2$

Q. Find the slope of the line whose inclination is:
​(i) $0^{\circ }$

Q. The slope of the line which is inclined at an angle of ${75}^{\circ }$ with the $x-$axis in anticlockwise direction is

1. $2-\sqrt{3}$
2. $2+\sqrt{3}$
3. $2-2\sqrt{3}$
4. $2+2\sqrt{3}$

Q. A line through $P(3,4)$ cuts the lines $x=6$ and $y=8$ at $L$ and $M$ respectively. $Q$ is a variable point on the line such that $\frac{1}{PQ}=\frac{1}{PL}+\frac{1}{PM}$ then the locus of $Q$ is

1. $4x+3y-36=0$
2. $x^2+y^2=36$
3. $3x-4y-36=0$
4. $4x^2-9y^2=36$

Q. The shortest distance between the pair of skew lines:
$x=-y+2=-z+2$ and $x-2=-y+1=z+1$ is $\frac{1}{\sqrt{n}}$. Then the value of $n$ is:

Q. The length (in units) of the perpendicular from the point $(-1,3,9)$ to the line $\frac{x-13}{5}=\frac{y+8}{-8}=\frac{z-31}{1}$ is:

1. $21$
2. $22$
3. $20$
4. $\sqrt{439}$

Q.

A square of side "a" lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an angle α where 0<α<90with the positive direction of x - axis. The equation of its diagonal not passing through the origin is

1. 

2. 

3. 

4. 

Q. A straight line L through the point $(3,-2)$ is inclined at an angle $60$ to the line $\sqrt{3}x+y=1$. If L also intersects the x-axis, the equation of L is-

1. $\sqrt{3}y+x-3+2\sqrt{3}=0$
2. $\sqrt{3}y-x+3+2\sqrt{3}=0$
3. $y+\sqrt{3}x+2-3\sqrt{3}=0$
4. $y-\sqrt{3}x+2+3\sqrt{3}=0$

Q. If $y=x^2+\frac{1}{x^2+\frac{1}{x^2+\frac{1}{x^2+...\infty }}}$, then $\frac{dy}{dx}$ is equal to

1. $\frac{xy}{y+x^2}$
2. $\frac{2xy}{2y-x^2}$
3. $\frac{xy}{y-x^2}$
4. $\frac{2x}{2+\frac{x^2}{y}}$

Q. If the line $y=x$ cuts the curve $y=2x^3+6x^2+x-4$ at three points $A,B$ and $C$. Then the value of $|OA.OB.OC|$ with $O$ being the origin is

1. $\sqrt{2}$
2. $2\sqrt{2}$
3. $4\sqrt{2}$
4. $8\sqrt{2}$

Q. The slope of the line formed by joining the points A(1, 3) and B(4, 7) is .

1. $\frac{4}{3}$
2. $\frac{-4}{3}$
3. $\frac{3}{4}$
4. $\frac{-3}{4}$